Exhaustion by compact sets

In mathematics, especially analysis, exhaustion by compact sets of a topological space $X$ is a nested sequence of compact subsets $K_{i}$ of $X$ (i.e. $K_{1}\subseteq K_{2}\subseteq K_{3}\subseteq \cdots$), such that $K_{i}$ is contained in the interior of $K_{i+1}$ , i.e. $K_{i}\subset {\text{int}}(K_{i+1})$ for each $i$ and $X=\bigcup _{i=1}^{\infty }K_{i}$ .

For example, consider $X={\mathbb {R} }^{n}$ and the sequence of closed balls $K_{i}=\{x:|x|\leq i\}$.

See also

• σ-compact space

References

• Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. ISBN 0-8218-1221-1.

• John Lee, Introduction to Topological Manifolds, Springer Verlag, 2nd ed. 2011. ISBN 978-1441979391.

• Hans Grauert and Reinhold Remmert, Theory of Stein Spaces, Springer Verlag (Classics in Mathematics), 2004. ISBN 978-3540003731.

External links

• "Exhaustion by compact sets". PlanetMath.